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Properties

Label 21.21.5043453350...1024.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{38}\cdot 809^{8}$
Root discriminant $44.93$
Ramified primes $2, 809$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\PSL(2,7)$ (as 21T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, -6, 334, -114, -4872, 1118, 23447, -6223, -49437, 14997, 52175, -17071, -29145, 9901, 8673, -2985, -1353, 465, 103, -35, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 35*x^19 + 103*x^18 + 465*x^17 - 1353*x^16 - 2985*x^15 + 8673*x^14 + 9901*x^13 - 29145*x^12 - 17071*x^11 + 52175*x^10 + 14997*x^9 - 49437*x^8 - 6223*x^7 + 23447*x^6 + 1118*x^5 - 4872*x^4 - 114*x^3 + 334*x^2 - 6*x - 2)
 
gp: K = bnfinit(x^21 - 3*x^20 - 35*x^19 + 103*x^18 + 465*x^17 - 1353*x^16 - 2985*x^15 + 8673*x^14 + 9901*x^13 - 29145*x^12 - 17071*x^11 + 52175*x^10 + 14997*x^9 - 49437*x^8 - 6223*x^7 + 23447*x^6 + 1118*x^5 - 4872*x^4 - 114*x^3 + 334*x^2 - 6*x - 2, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 35 x^{19} + 103 x^{18} + 465 x^{17} - 1353 x^{16} - 2985 x^{15} + 8673 x^{14} + 9901 x^{13} - 29145 x^{12} - 17071 x^{11} + 52175 x^{10} + 14997 x^{9} - 49437 x^{8} - 6223 x^{7} + 23447 x^{6} + 1118 x^{5} - 4872 x^{4} - 114 x^{3} + 334 x^{2} - 6 x - 2 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(50434533500374375555123632117121024=2^{38}\cdot 809^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.93$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 809$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} + \frac{1}{4} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{15} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{8} a^{9} - \frac{1}{2} a^{8} - \frac{3}{8} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{14} + \frac{1}{4} a^{11} - \frac{1}{8} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{6472} a^{19} + \frac{141}{6472} a^{18} - \frac{143}{3236} a^{17} - \frac{239}{6472} a^{16} + \frac{563}{6472} a^{15} - \frac{98}{809} a^{14} + \frac{43}{3236} a^{13} - \frac{473}{3236} a^{12} - \frac{2611}{6472} a^{11} - \frac{2341}{6472} a^{10} - \frac{703}{3236} a^{9} - \frac{689}{6472} a^{8} - \frac{469}{6472} a^{7} + \frac{591}{3236} a^{6} + \frac{609}{1618} a^{5} + \frac{1389}{3236} a^{4} + \frac{19}{809} a^{3} + \frac{751}{3236} a^{2} - \frac{1129}{3236} a + \frac{771}{1618}$, $\frac{1}{3569412670320173841184} a^{20} + \frac{24131607872250341}{446176583790021730148} a^{19} + \frac{174607700581853606405}{3569412670320173841184} a^{18} - \frac{95992739439246598687}{1784706335160086920592} a^{17} + \frac{92839092186182575547}{3569412670320173841184} a^{16} + \frac{53885977486993769517}{892353167580043460296} a^{15} + \frac{127040471630325323359}{3569412670320173841184} a^{14} - \frac{153294972228819869181}{1784706335160086920592} a^{13} + \frac{603401178813590394591}{3569412670320173841184} a^{12} - \frac{408437144386700734155}{892353167580043460296} a^{11} + \frac{1124240507247422074901}{3569412670320173841184} a^{10} - \frac{806934579762906122839}{1784706335160086920592} a^{9} - \frac{329768406825211489977}{3569412670320173841184} a^{8} - \frac{416125693715545309581}{892353167580043460296} a^{7} + \frac{23821068675037713353}{3569412670320173841184} a^{6} - \frac{266900379485174548295}{1784706335160086920592} a^{5} + \frac{242101215887381773551}{892353167580043460296} a^{4} - \frac{143795894569624343903}{892353167580043460296} a^{3} - \frac{28471775953665843951}{1784706335160086920592} a^{2} - \frac{352268468812901721725}{892353167580043460296} a + \frac{236333156769368747163}{1784706335160086920592}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $20$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 232658701418 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PSL(2,7)$ (as 21T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 168
The 6 conjugacy class representatives for $\PSL(2,7)$
Character table for $\PSL(2,7)$

Intermediate fields

7.7.670188544.2, 7.7.670188544.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 7 siblings: 7.7.670188544.2, 7.7.670188544.1
Degree 8 sibling: 8.8.28072042781802496.1
Degree 14 siblings: 14.14.18813561479041924489805824.1, 14.14.18813561479041924489805824.4
Degree 24 sibling: data not computed
Degree 28 sibling: data not computed
Degree 42 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.10.2$x^{6} + 2 x^{5} + 2 x^{4} + 2 x^{2} + 2$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
2.12.26.65$x^{12} + 4 x^{3} + 2$$12$$1$$26$$S_4$$[8/3, 8/3]_{3}^{2}$
809Data not computed